derivative Definition and Topics - 137 Discussions

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.
Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.
The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.

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  1. B

    Derivative Question

    Homework Statement [/B] I've tried to search this up but to no avail. How am I suppose to solve this: d2y/dx3 Homework Equations N/A The Attempt at a Solution Here's what I think I need to do: 1: Square and cube y and x respectively. 2: Find the second and third derivative of y and x...
  2. Rectifier

    Integration with variable substitution

    Hello, I am having trouble with solving the problem below The problem Find all primitive functions to ## f(x) = \frac{1}{\sqrt{a+x^2}} ##. (Translated to English) The attempt I am starting with substituting ## t= \sqrt{a+x^2} \Rightarrow x = \sqrt{t^2 - a} ## in $$ \int \frac{1}{\sqrt{a+x^2}}...
  3. W

    Derivative in spherical coordinates

    Homework Statement -here is the problem statement -here is a bit of their answer Homework Equations Chain rule, partial derivative in spherical coord. The Attempt at a Solution I tried dragging out the constant and partial derivate with respect to t but still I can't reach their df/dt and...
  4. O

    A Frechet v Gateaux Derivative and the calculus of variations

    Good Morning Could someone please distinguish between the Frechet and Gateaux Derivatives and why one is better to use in the Calculus of Variations? In your response -- if you are so inclined -- please try to avoid the theoretical foundations of this distinction (as I can investigate that by...
  5. Adeel Ahmad

    Partial Derivatives

    Homework Statement So I know I have to take the derivative with respect to x, then respect to y, then respect to z, but I am not getting the right answer. I know that the answer is 0 and my professor did it with very few steps that I do not understand. Can someone please guide me through it?
  6. Kanashii

    Finding the second derivative using central difference formula

    Homework Statement Develop aprogram that will determine the second derivative of pi(16 x^2 - y^4) at y=2 with step sizes of 0.1, 0.01, 0.001…. until the absolute error (numerical-analytical) converges to 0.00001. Use the 2nd order Central Difference Formula. User Input: y, tolerance Output: h...
  7. MiLara

    I Why do some but not all derivatives have physical meaning?

    I know that taking the derivative of certain functions that explain physical phenomena can lead to another statement describing the physical system, the most famous being the derivatives of position. That is, position-->velocity-->acceleration-->jerk-->jounce...and taking any other further...
  8. M

    Theoretic doubt about the definition of derivatives.

    Homework Statement Hi, this is a question that has been bothering me for a while. (Im in calculus II at the moment) Why do i need to derivate some functions by definition and other times i dont? for example if somebody asks me to calculate the partial derivatives of a branch function in a a...
  9. M

    I A-level differentiation/derivative dilemma

    Hello, and thank you for your time. I just started my A-levels derivatives/differentiation , and I would be more than happy if you could help me clarify it. For example I know that y is a function in terms of x right? y=f(x) The derivative of it is f'(x)=dy/dx . This means it is the rate of...
  10. CivilSigma

    Derivative in mass flow rate equation - Hydrology

    Hello, I am working with the mass flow rate equation which is: $$\frac{d \dot{m}}{dt}=\dot{m}_{in}-\dot{m}_{out}$$ To determine the change of the height of water in a reservoir. Assuming m_in = 10 and m_out = sqrt(20h), then : $$\frac{d (\rho \cdot Q) }{dt}=\rho \cdot Q_{in} - \rho\cdot...
  11. orion

    I Need help with derivative notation

    If I have a scalar function of a variable ##x## I can write the derivative as: ##f'(x)=\frac{df}{dx}##. Now suppose ##x## is no longer a single variable but a vector: ## x=(x^1, x^2, ..., x^n)##. Then of course we have for the derivative ##(\frac{\partial f}{\partial x^1}, ..., \frac{\partial...
  12. Jeffack

    A Create function which meets slope and point requirements

    I am trying to create a function of A and x which has the following properties. A is a scaling parameter that determines the shape of the function. I write the function below in f(A,x) form 1) f(A,1)=1 always 2) For all x>1, 0<f ' (x)<1 3) As A approaches some upper bound (which could be...
  13. redtree

    A Relationship between metric tensor and position vector

    Given the definition of the covariant basis (##Z_{i}##) as follows: $$Z_{i} = \frac{\delta \textbf{R}}{\delta Z^{i}}$$ Then, the derivative of the covariant basis is as follows: $$\frac{\delta Z_{i}}{\delta Z^{j}} = \frac{\delta^2 \textbf{R}}{\delta Z^{i} \delta Z^{j}}$$ Which is also equal...
  14. DavideGenoa

    I Substitution in a Lebesgue integral

    Hi, friends! I read that, if ##f\in L^1[c,d]## is a Lebesgue summable function on ##[a,b]## and ##g:[a,b]\to[c,d]## is a differomorphism (would it be enough for ##g## to be invertible and such that ##g\in C^1[a,b]## and ##g^{-1}\in C^1[a,b]##, then...
  15. MrDickinson

    Need a little help with this related rates problem

    Can someone help me with this? (dA/dt)=1cm/s (cm^2 whatever...leave out trivial corrections). A=pir^2 (dA/dt)=2pir(dR/dt) Multiply through by (1/2pir) (dA/dt)/(2pir)=dR/Dt What is the rate of change of the radius for a circumfrance of 2 I just used the related rates formula that I derived for...
  16. funlord

    I I can't understand the fact

    I can's understand the fact about the equation i cant prove the equation from the first attachment to the second attachment pls help. Sorry for bad english
  17. E

    I How to Convert this Derivative?

    I have a derivative of a function with respect to ##\log \left(r\right)##: \begin{equation*} \frac{dN\left(r\right)}{d \log\left(r\right)} = \frac{N}{\sqrt{2\pi} \log\left(\sigma\right)} \exp\left\{-\frac{\left[\log \left(r\right) - \log\left(r_M\right)\right]^2}{2...
  18. D

    I What is the derivative of a matrix transpose?

    Hi! As the title says, what is the derivative of a matrix transpose? I am attempting to take the derivative of \dot{q} and \dot{p} with respect to p and q (on each one). Any advice?
  19. AwesomeTrains

    Ladder operator commutator with arbitary function

    Hey there! 1. Homework Statement I've been given the operators a=\sqrt\frac{mw}{2\hbar}x+i\frac{p}{\sqrt{2m\hbar w}} and a^\dagger=\sqrt\frac{mw}{2\hbar}x-i\frac{p}{\sqrt{2m\hbar w}} without the constants and definition of the momentum operator: a=x+\partial_x and a^\dagger=x-\partial_x with...
  20. D

    I How to understand the notion of a limit of a function

    I am trying to explain to someone the formal notion of a limit of a function, however it has made me realise that I might have some faults in my own understanding. I will write down how I understand the subject and would very much appreciate if someone(s) can point out any...
  21. K

    IalChange of variables/verifying solution

    Homework Statement Trying to use change of variables to simplify the schrodinger equation. I'm clearly going wrong somewhere, but can't see where. Homework Equations [/B] Radial Schrodinger: -((hbar)2)/2M * [(1/r)(rψ)'' - l(l+1)/(r^2) ψ] - α(hbar)c/r ψ = Eψ The Attempt at a Solution...
  22. StanEvans

    I Magnitude of the Second Derivative

    So to find the x values of the stationary points on the curve: f(x)=x3+3x2 you make f '(x)=0 so: 3x2+6x=0 x=0 or x=-2 Then to find which of these points are maximum or minimum you do f ''(0) and f ''(-2) so: 6(0)+6=6 6(-2)+6=-6 so the maximum has an x value of -2 and the minimum has an x value...
  23. Jonathan Densil

    Uncertainty of Derivatives

    Homework Statement I know this is more of a physics question, but I tried there and wasn't successful. I have done a physics experiment measuring the weight as a function time of the discharge of water from a cylindrical bottle with a pinhole at the bottom. What I ultimately want to get at is...
  24. kevin2016

    A What is the closed-form solution using ALS algorithm to optimize

    C \in \mathbb{R}^{m \times n}, X \in \mathbb{R}^{m \times n}, W \in \mathbb{R}^{m \times k}, H \in \mathbb{R}^{n \times k}, S \in \mathbb{R}^{m \times m}, P \in \mathbb{R}^{n \times n} ##{S}## and ##{P}## are similarity matrices (symmetric). ##\lambda##, ##\alpha## and ##\beta## are...
  25. ZKhawla

    Dirac derivative and signal energy distribution

    Hi, I'm writing a mathematical expression of energy distribution of a signal, and in the formula I’ve found first and second derivative of delta function. I have to analyze my result but couldn’t found how to read these two derivative from an energy point of view. And how can we see further...
  26. G

    Find the derivative of the function(Quotient rule)

    Homework Statement Find the derivative of the function y = (3-2x^3+x^6 )/x^9 Homework Equations Derivatives The Attempt at a Solution I have tried to use the quotient rule and got to -6x^11 + 6x^14 - 27x^8 + 18x ^24 - 9x ^14 / (x^9)^2 Which doesn't look close to the answer -27/x^10 +...
  27. G

    Finding the total distance traveled by the body at interval

    Homework Statement At time t, the position of a body moving along the s-axis is s= t^3 -12t^2 + 36t m(meters) Find the total distance traveled by the body from t = 0 to t = 3. Homework Equations Derivatives The Attempt at a Solution I got the derivative which is 3t^2 - 24t + 36(meters) I...
  28. Rmehtany

    A nasty integral to compute

    Hey Guys! I was working on an integration problem, and I "simplified" the integral to the following: $$\int \limits_0^{2\pi} (7.625+.275 \cos(4x))^{1.5} \cdot (A \cos(Nx) + B \sin(Nx)) \cdot (Z-v \cos(x)) dx$$ This integral may seem impossible (I have almost lost all hope on doing this...
  29. M

    Is ln(x) differentiable at negative x-axis

    Since lnx is defined for positive x only shouldnt the derivative of lnx be 1/x, where x is positive. My books does not specify that x must be positive, so is lnx differentiable for all x?
  30. J

    Implications of varying the definition of the derivative?

    I have been playing around with calculus for a while and I wondered what would it be like to make some changes to the definition of derivatives. I'd like to look at the original definition of derivatives in this way (everything is in lim Δx→0): F(x+Δx) - F(x) = F'(x) * Δx The Δx factor...
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